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Units

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An element of a set is a unit if its multiplicative inverse belongs to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm 1 and are thus complex roots of unity.

Units in quadratic integer rings

Imaginary quadratic integer rings have only a few units (see: Quadratic integer rings§Theorem I1). The primitive root of unity of degree 3

ω := ei  2 π / 3 = −
1
2
  +
2  −3 
2
 ,

is used in the following table.

Units of imaginary quadratic integer rings
d
Units
  ≤    − 5
{( − 1) 0, ( − 1) 1} = {1,  − 1}
 − 3
{ω 0, ω 1, ω 2, ω 3, ω 4, ω 5} = {1, ω, ω 2,  − 1,  − ω,  − ω 2}
 − 2
{( − 1) 0, ( − 1) 1} = {1,  − 1}
 − 1
{i 0, i 1, i 2, i 3} = {1, i,  − 1,  −  i}

Real quadratic integer rings have infinitely many units (see: Quadratic integer rings§Theorem R1), which are all powers of each other, some multiplied by 1. For that reason, the table below can’t be complete like the table above.

An inefficient way to find a unit of a real quadratic integer ring
ℤ [
2  d
 ]
other than 1 or  − 1 is to try each positive value of
b
starting with 1 and going up until
2  1 + db 2
is an integer.
Units of real quadratic integer rings
d
Units
2
1 +
2  2 
3
2 +
2  3 
4 N/A
5
1
2
  +
2  5 
2
 (golden ratio)
6
5 + 2 
2  6 
7
8 + 3 
2  7 
8 N/A, but note that
3 2  −  8  (1 2) = 1
9 N/A
10
3 +
2  10 
11
10 + 3 
2  11 
12 N/A, but
7 2  −  12  (2 2) = 1
13
3
2
  +
2  13 
2
14
15 + 4 
2  14 
15
4 +
2  15 
16 N/A
17
4 +
2  17 
        
d
Units
18 N/A, but
17 2  −  18  (4 2) = 1
19
170 + 39 
2  19 
20 N/A, but
9 2  −  20  (2 2) = 1
21
5
2
  +
2  21 
2
22
197 + 42 
2  22 
23
24 + 5 
2  23 
24 N/A, but
5 2  −  24  (1 2) = 1
25 N/A
26
5 +
2  26 
27 N/A, but
26 2  −  27  (5 2) = 1
28 N/A, but
127 2  −  28  (24 2) = 1
29
5
2
  +
2  29 
2
30
11 + 2 
2  30 
31
1520 + 273 
2  31 
32 N/A, but
17 2  −  32  (3 2) = 1
33
23
2
  +
4 
2  33 
2

Examples

  • The units of
    are
    {1 0} = {1}
    , where
    1   ≡   ei 2 π
    . (See natural numbers.)
  • The units of
    are
    {( − 1) 0, ( − 1) 1} = {1,  − 1}
    , where
     − 1   ≡   ei  π
    . (See rational integers.)
  • The units of
    ℤ [ω]
    are
    {ω 0, ω 1, ω 2} = {1, ω, ω 2}
    , where
    ω   ≡   ei  2 π / 3
    . (See Eisenstein integers.)
  • The units of
    ℤ [i]
    are
    {i 0, i 1, i 2, i 3} = {1, i,  − 1,  −  i}
    , where
    i   ≡   ei  π / 2
    . (See Gaussian integers.)
  • The units of
    are
    ℚ ∖{0}
    .
  • The units of
    are
    ℝ ∖{0}
    .
  • The units of
    are
    ℂ ∖{0}
    .

See also